Modulation Equation for SPDEs in unbounded domains with space-time white noise -- Linear Theory

TL;DR

This paper develops a linear theory for modulation equations approximating SPDEs on unbounded domains with space-time white noise, focusing on the Swift-Hohenberg equation to facilitate stochastic approximation analysis.

Contribution

It introduces a linear framework for modulation equations in unbounded domains, addressing the challenges posed by space-time white noise and setting the stage for nonlinear extensions.

Findings

Established linear approximation results for SPDEs on unbounded domains.

Analyzed the impact of space-time white noise's translation invariance on error estimates.

Provided technical tools for future nonlinear stochastic modulation theory.

Abstract

We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation. As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift-Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.